Catch up?
Yet another I-am-sorry-I-have-not-blogged-in-a-while”post. Will try to catch up on my past course material. Next semester seems pretty busy, at least in terms of coursework. It is now or never…
Group theory – Week 07
For our purposes, a Lie algebra is defined as a linear vector space 
![\left[ , \right]](http://l.wordpress.com/latex.php?latex=%5Cleft%5B+%2C+%5Cright%5D&bg=fff&fg=222&s=0 "\left[ , \right]")
- closure: For
- antisymmetry:
- linearity of the bracket:
- associativity:
![X,Y \in L, [X , Y] \in L](http://l.wordpress.com/latex.php?latex=X%2CY+%5Cin+L%2C+%5BX+%2C+Y%5D+%5Cin+L&bg=fff&fg=222&s=0 "X,Y \in L, [X , Y] \in L")
![\left[X, Y\right] = - \left[Y, X\right]](http://l.wordpress.com/latex.php?latex=%5Cleft%5BX%2C+Y%5Cright%5D+%3D+-+%5Cleft%5BY%2C+X%5Cright%5D&bg=fff&fg=222&s=0 "\left[X, Y\right] = - \left[Y, X\right]")
![\left[aX + bY , Z\right] = a \left[X , Z\right] + b\left[Y , Z\right]](http://l.wordpress.com/latex.php?latex=%5Cleft%5BaX+%2B+bY+%2C+Z%5Cright%5D+%3D+a+%5Cleft%5BX+%2C+Z%5Cright%5D+%2B+b%5Cleft%5BY+%2C+Z%5Cright%5D&bg=fff&fg=222&s=0 "\left[aX + bY , Z\right] = a \left[X , Z\right] + b\left[Y , Z\right]")
![\left[X, \left[Y, Z\right]\right] +\left[Z, \left[X, Y\right]\right] + \left[Y, \left[Z, X\right]\right]= 0](http://l.wordpress.com/latex.php?latex=%5Cleft%5BX%2C+%5Cleft%5BY%2C+Z%5Cright%5D%5Cright%5D+%2B%5Cleft%5BZ%2C+%5Cleft%5BX%2C+Y%5Cright%5D%5Cright%5D+%2B+%5Cleft%5BY%2C+%5Cleft%5BZ%2C+X%5Cright%5D%5Cright%5D%3D+0&bg=fff&fg=222&s=0 "\left[X, \left[Y, Z\right]\right] +\left[Z, \left[X, Y\right]\right] + \left[Y, \left[Z, X\right]\right]= 0")
A vector in 
![\left[T_{a}, T_{b}\right] = f_{ab}^{c} T_{c}.](http://l.wordpress.com/latex.php?latex=%5Cleft%5BT_%7Ba%7D%2C+T_%7Bb%7D%5Cright%5D+%3D+f_%7Bab%7D%5E%7Bc%7D+T_%7Bc%7D.&bg=fff&fg=222&s=0 "\left[T_{a}, T_{b}\right] = f_{ab}^{c} T_{c}.")
We will be considering real forms, where the structure constants are always real numbers.
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